Daylight hours


The daylight hours are maximum possible duration of sunshine for a given day of the year (Allen et al. 1998).


Formulation according to FAO:

The daylight hours \(N\) are obtained as follows (Allen et al. 1998):


where \(\omega_s\) is the sunset hour angle [rad], which is calculated as follows:

\[\omega_s=\arccos [-\tan\varphi\cdot\tan\delta]\]

where \(\delta\) is the solar declination [rad], and \(\varphi\) the latitude [rad].

The solar declination \(\delta\) is calculated as follows:

\[\delta=0.409\cdot\sin\bigg(\dfrac{2\pi}{365}\cdot J-1.39\bigg)\]

where \(J\) is the number of the day in the year between 1 (1 January) and 365 or 366 (31 December).

The conversion from decimal degrees to radians is obtained as follows:

\[\mbox{[Radians]}=\dfrac{\pi}{180} \mbox{ [decimal degrees]}\]

NB: for calculating potential evapotranspiration according to Thornthwaite, a daylight coefficient \(C\) is required, which corresponds to the duration of sunlight in units of 12 hours. This daylight coefficient \(C\) is thus given by:


Formulation according to NFDRS

The procedure for determining daylight hours \(N_{nfdrs}\) proposed by (Cohen & Deeming (1985) in order to calculate some components of the NFDRS is slightly different from that proposed by (Allen et al. (1998):


where \(\varphi\) is latitude [rad] and \(\delta\) the solar declination [rad].

The latitude \(\varphi\) [rad] is given by:


where \(\varphi_{deg}\) is the latitude in decimal degrees.

The solar declination \(\delta\) is calculated as follows:


where \(J\) is the Julian date.


Cohen & Deeming (1985)
Allen et al. (1998)


Variable Description Unit
\(T\) air temperature °C
\(T_{dew}\) dew point temperature °C
\(H\) air humidity %
\(P\) rainfall mm
\(U\) windspeed m/s
\(w\) days since last rain
(or rain above threshold)
\(rr\) days with consecutive rain d
\(\Delta t\) time increment d
\(\Delta{e}\) vapor pressure deficit kPa
\(e_s\) saturation vapor pressure kPa
\(e_a\) actual vapor pressure kPa
\(p_{atm}\) atmospheric pressure kPa
\( PET\) potential evapotranspiration mm/d
\(r\) soil water reserve mm
\(r_s\) surface water reserve mm
\(EMC\) equilibrium moisture content %
\(DF\) drought factor -
\(N\) daylight hours hr
\(D\) weighted 24-hr average moisture condition hr
\(\omega\) sunset hour angle rad
\(\delta\) solar declination rad
\(\varphi\) latitude rad
\(Cc\) cloud cover Okta
\(J\) day of the year (1..365/366) -
\(I\) heat index -
\(R_n\) net radiation MJ⋅m-2⋅d-1
\(R_a\) daily extraterrestrial radiation MJ⋅m-2⋅d-1
\(R_s\) solar radiation MJ⋅m-2⋅d-1
\(R_{so}\) clear-sky solar radiation MJ⋅m-2⋅d-1
\(R_{ns}\) net shortwave radiation MJ⋅m-2⋅d-1
\(R_{nl}\) net longwave radiation MJ⋅m-2⋅d-1
\(\lambda\) latent heat of vaporization MJ/kg
\(z\) elevation m a.s.l.
\(d_r\) inverse relative distance Earth-Sun -
\(\alpha\) albedo or canopy reflection coefficient -
\(\Delta\) slope of the saturation vapor pressure curve kPa/°C
\(Cc\) cloud cover eights
\(ROS\) rate of spread m/h
\(RSF\) rate of spread factor -
\(WF\) wind factor -
\(WRF\) water reserve factor -
\(FH\) false relative humidity -
\(FAF\) fuel availability factor -
\(PC\) phenological coefficient -

Suffix Description
\(-\) mean / daily value
\(_{max}\) maximum value
\(_{min}\) minimum value
\(_{12}\) value at 12:00
\(_{13}\) value at 13:00
\(_{15}\) value at 15:00
\(_{m}\) montly value
\(_{y}\) yearly value
\(_{f/a}\) value at fuel-atmosphere interface
\(_{dur}\) duration
\(_{soil}\) value at soil level

Constant Description
\(e\) Euler's number
\(\gamma\)psychrometric constant
\(G_{SC}\)solar constant
\(\sigma\)Stefan-Bolzmann constant